Problem: Solve the equation $|y-6| + 2y = 9$ for $y$.
Solution: We consider two cases, $y\ge 6$ and $y < 6$.

Case 1: $y \ge 6:$  If $y \ge 6$, then $|y-6| = y-6$ and our equation is $y-6+2y=9$.  So, we have $3y = 15$, or $y=5$. However, $y=5$ does not satisfy $y\ge 6$.  Testing $y=5$, we have $|5-6| + 2\cdot 5 =11$, not 9, and we see that $y=5$ is not a solution.

Case 2: $y < 6:$   If $y<6$, then $|y-6| = -(y-6) = -y+6$, so our equation is $-y+6+2y = 9$, from which we have $y=\boxed{3}$. This is a valid solution, since $y=3$ satisfies the restriction $y<6$.